Chapter 2.1, Problem 93AYU

a)

To determine

To identify: the dependent and the independent variables for the given function.

Answer to Problem 93AYU

The dependent and the independent variables for the given function are P and a respectively.

Explanation of Solution

Given information:

A population function is given as

  $P\left(a\right)=0.015a^2-4.962a+290.580$

Here $P$ is in millions and age of Americans are a years or older.

Concept used:

Consider two sets A and B . If each element $x$ in the set A has exactly one element $y$ in the set B, the relation between them is known as a function. The ordered pair is $\left(x,y\right)$ .

The values of first coordinate $x$ is known as input values and the values of second coordinate $y$ is known as output values of a function.

For each random value of $x$ , the variable $y$ will give a different value.

So, value of $y$ is dependent on each value of $x$ .

Calculation:

Consider the given function.

  $P\left(a\right)=0.015a^2-4.962a+290.580$

Here, for every input of age a , population P has a specific output.

Therefore, for the equation $P\left(a\right)=0.015a^2-4.962a+290.580$ , age $a$ is independent variable and population $P$ is dependent variable.

b)

To determine

To find: the value of $P\left(20\right)$ and interpret the result.

Answer to Problem 93AYU

  $P\left(20\right)=197.34$ million.

  $P\left(20\right)=197.34$ million means in 2005 there was 197.34 million people of age 20 or older.

Explanation of Solution

Given information:

A population function is given as

  $P\left(a\right)=0.015a^2-4.962a+290.580$

Here $P$ is in millions and age of Americans are a years or older.

Calculation:

Consider the given function.

  $P\left(a\right)=0.015a^2-4.962a+290.580$

For evaluating $P\left(20\right)$ , substitute 20 for a in above equation.

  $\begin{array}{c}P\left(20\right)=0.015{\left(20\right)}^2-4.962\left(20\right)+290.580\\ =197.34\end{array}$

Here $P$ is in millions and age of Americans are a years or older.

  $P\left(20\right)=197.34$ million means in 2005 there was 197.34 million people of age 20 or older.

c)

To determine

To find: the value of $P\left(0\right)$ and interpret the result.

Answer to Problem 93AYU

  $P\left(0\right)=290.58$ million.

  $P\left(0\right)=290.58$ million means in 2005 there was 290.58 million people of age 0 or older.

Explanation of Solution

Given information:

A population function is given as

  $P\left(a\right)=0.015a^2-4.962a+290.580$

Here $P$ is in millions and age of Americans are a years or older.

Calculation:

Consider the given function.

  $P\left(a\right)=0.015a^2-4.962a+290.580$

For evaluating $P\left(0\right)$ , substitute 0 for a in above equation.

  $\begin{array}{c}P\left(0\right)=0.015{\left(0\right)}^2-4.962\left(0\right)+290.580\\ =290.58\end{array}$

Here $P$ is in millions and age of Americans are a years or older.

  $P\left(0\right)=290.58$ million means in 2005 there was 290.58 million people of age 0 or older.